Quantum gates with hot trapped ions
合集下载
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
In order to analyze the scheme, we consider two ions confined in a linear trap [8]. We will assume that the motion of the ions is frozen with respect to the y and z axis. The Hamiltonian describing this situation is
(February 9, 2008)
arXiv:quant-ph/9712012v1 4 Dec 1997
We propose a scheme to perform a fundamental two–qubit gate between two trapped ions using ideas from atom interferometry. As opposed to the scheme considered by J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995), it does not require laser cooling to the motional
+
p22 2m
+
V
(x1)
+
V
(x2)
+
e2 4πǫ0|x1 −
x2| ,
(2)
where V denotes the external confining potential along the x axis, which we assume to be symmetric V (x) = V (−x). We denote by xe the separation of the ions at the equilibrium position. Using the center–off–mass and the relative coordinates,
Quantum computation allows the development of polynomial-time algorithms for computational problems such a prime factoring, which have previously been viewed as intractable [1]. This has motivated studies into the feasibility of actual implementation of quantum computers in physical systems [2]. The task of designing a QC is equivalent to finding a physical realization of quantum gates between a set of qubits, where a qubit refers to a two–level system {|0 , |1 }. Any operation can be decomposed into rotations on a single qubit and a universal two-bit gate, e.g. a gate defined by
xc = (x1 + x2)/2, xr = x1 − x2 − xe, pc = p1 + p2, pr = (p1 − p2)/2,
(3a) (3b)
and expanding around the equilibrium position xc = xr = 0 up to second order, we obtain
Cˆ12 : |ǫ1 |ǫ2 → |ǫ1 |(1 + ǫ1) ⊕ ǫ2 (ǫ1,2 = 0, 1). (1)
Implementation of a quantum computer requires precise control of the Hamiltonian operations and a high degree of coherence. Achieving the conditions for quantum computation is extremely demanding, and only a few systems have been identified as possible candidates to build small scale models in the lab [1–4]. One of the most promising examples is a string of cold ions stored in a linear trap [3]. In this ion trap quantum computer qubits are stored in long–lived internal atomic ground states. Single bit operations in this system are accomplished by directing different laser beams to each of the ions, and a fundamental gate operation is implemented by exciting the collective quantized motion of the ions with lasers, where the exchange of phonons serves as a data bus to transfer quantum information between the qubits. Prospects of building small ion trap quantum computers in the near future are good, but the question remains whether stringent requirements in implementing two bit quantum gates can be relaxed: while decoherence of the qubit stored in the atomic ground state is not an issue, cooling of ions to the vibrational ground state to prepare a pure initial state for the collective phonon mode remains a challenge [5]. The question arises whether quantum gates can be performed starting from thermal or mixed states of phonon modes. The task of implementing quantum computing in ”hot” systems seems particularly timely in view of the recent
H = Hho + Vcor(xc, xr),
(4)
1
where
Hho
=
p2c 2mc
+
p2r 2mr
+
1 2
interest in NMR quantum computing [4], where quantum states are stored as pseudo-pure states in an ensemble of ”hot” spins.
In this Letter we will discuss the implementation of a universal two–bit gate between two ions in a linear trap with the phonon modes initially in a thermal (mixed) state. The novel concept behind the gate operation is to implement conditional dynamics based on atom interferometry of two entangled atoms [6,7]. The two-bit gate operation proceeds as follows: ion 2 is kicked left or right depending on its internal state with laser light (see Fig. 1(a) at t = 0). Thus the ion 1 will experience a kick via the Coulomb repulsion conditional to the internal state of the first ion. The corresponding wave packet will evolve into a superposition of two spatial wavepackets which are entangled to the internal state of the control ion. Provided the spatial splitting of the wave packet of the second ion is sufficiently large (at time t0 in Fig. 1(a)), we can manipulate the internal state of the target atom depending on its spatial position, i.e depending on the state of the first ion, and thus implement a gate operation on the qubits. With time these atomic wave packets will oscillate in the trap, and with a proper sequence of laser pulses this momentum transfered to the two atoms can be undone to restore the original motional state of the ion (Fig. 1(a) at time tg). The motional state of the atom will thus factorize from the internal atomic state before and after the gate, independent of being a mixed or a pure state.
Quantum gates with ”hot” trapped ions
J. F. Poyatos(1,2), J.I. Cirac(1,2), and P. Zoller(2) (1) Departamento de F´ısica Aplicada, Universidad de Castilla–La Mancha, 13071 Ciudad Real, Spain (2) Institut fu¨r Theoretische Physik, Universit¨at Innsbruck, Technikerstrasse 25, A–6020 Innsbruck, Austria.
(February 9, 2008)
arXiv:quant-ph/9712012v1 4 Dec 1997
We propose a scheme to perform a fundamental two–qubit gate between two trapped ions using ideas from atom interferometry. As opposed to the scheme considered by J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995), it does not require laser cooling to the motional
+
p22 2m
+
V
(x1)
+
V
(x2)
+
e2 4πǫ0|x1 −
x2| ,
(2)
where V denotes the external confining potential along the x axis, which we assume to be symmetric V (x) = V (−x). We denote by xe the separation of the ions at the equilibrium position. Using the center–off–mass and the relative coordinates,
Quantum computation allows the development of polynomial-time algorithms for computational problems such a prime factoring, which have previously been viewed as intractable [1]. This has motivated studies into the feasibility of actual implementation of quantum computers in physical systems [2]. The task of designing a QC is equivalent to finding a physical realization of quantum gates between a set of qubits, where a qubit refers to a two–level system {|0 , |1 }. Any operation can be decomposed into rotations on a single qubit and a universal two-bit gate, e.g. a gate defined by
xc = (x1 + x2)/2, xr = x1 − x2 − xe, pc = p1 + p2, pr = (p1 − p2)/2,
(3a) (3b)
and expanding around the equilibrium position xc = xr = 0 up to second order, we obtain
Cˆ12 : |ǫ1 |ǫ2 → |ǫ1 |(1 + ǫ1) ⊕ ǫ2 (ǫ1,2 = 0, 1). (1)
Implementation of a quantum computer requires precise control of the Hamiltonian operations and a high degree of coherence. Achieving the conditions for quantum computation is extremely demanding, and only a few systems have been identified as possible candidates to build small scale models in the lab [1–4]. One of the most promising examples is a string of cold ions stored in a linear trap [3]. In this ion trap quantum computer qubits are stored in long–lived internal atomic ground states. Single bit operations in this system are accomplished by directing different laser beams to each of the ions, and a fundamental gate operation is implemented by exciting the collective quantized motion of the ions with lasers, where the exchange of phonons serves as a data bus to transfer quantum information between the qubits. Prospects of building small ion trap quantum computers in the near future are good, but the question remains whether stringent requirements in implementing two bit quantum gates can be relaxed: while decoherence of the qubit stored in the atomic ground state is not an issue, cooling of ions to the vibrational ground state to prepare a pure initial state for the collective phonon mode remains a challenge [5]. The question arises whether quantum gates can be performed starting from thermal or mixed states of phonon modes. The task of implementing quantum computing in ”hot” systems seems particularly timely in view of the recent
H = Hho + Vcor(xc, xr),
(4)
1
where
Hho
=
p2c 2mc
+
p2r 2mr
+
1 2
interest in NMR quantum computing [4], where quantum states are stored as pseudo-pure states in an ensemble of ”hot” spins.
In this Letter we will discuss the implementation of a universal two–bit gate between two ions in a linear trap with the phonon modes initially in a thermal (mixed) state. The novel concept behind the gate operation is to implement conditional dynamics based on atom interferometry of two entangled atoms [6,7]. The two-bit gate operation proceeds as follows: ion 2 is kicked left or right depending on its internal state with laser light (see Fig. 1(a) at t = 0). Thus the ion 1 will experience a kick via the Coulomb repulsion conditional to the internal state of the first ion. The corresponding wave packet will evolve into a superposition of two spatial wavepackets which are entangled to the internal state of the control ion. Provided the spatial splitting of the wave packet of the second ion is sufficiently large (at time t0 in Fig. 1(a)), we can manipulate the internal state of the target atom depending on its spatial position, i.e depending on the state of the first ion, and thus implement a gate operation on the qubits. With time these atomic wave packets will oscillate in the trap, and with a proper sequence of laser pulses this momentum transfered to the two atoms can be undone to restore the original motional state of the ion (Fig. 1(a) at time tg). The motional state of the atom will thus factorize from the internal atomic state before and after the gate, independent of being a mixed or a pure state.
Quantum gates with ”hot” trapped ions
J. F. Poyatos(1,2), J.I. Cirac(1,2), and P. Zoller(2) (1) Departamento de F´ısica Aplicada, Universidad de Castilla–La Mancha, 13071 Ciudad Real, Spain (2) Institut fu¨r Theoretische Physik, Universit¨at Innsbruck, Technikerstrasse 25, A–6020 Innsbruck, Austria.